On Dedekind’s Explanation of the Finite in Terms of the Infinite Erich Reck University of California at Riverside Cambridge, September 22, 2013
2 Introduction • Prominent in discussions of the mathematical infinite is typically: Georg Cantor ; earlier also Bolzano , Galilei , all the way back to Aristotle . One highly influential authority: David Hilbert , with his well-known remark that • “no one shall drive us from the paradise that Cantor created for us” (1926). I want to highlight another seminal figure: Richard Dedekind . • A first remark guiding me is by Ernst Zermelo , who wrote that modern set theory • was “created by Cantor and Dedekind” (1908). Also Hilbert, who was fascinated by Dedekind’s and Frege’s attempts to “explain • the finite in terms of the infinite” (1922), even though he took them to have failed. Finally, cf. Akihiro Kanamori: the actual infinite first “entered [mainstream] • mathematics in Dedekind’s work” (2012), already in the 1850s. Claim: Dedekind was as important as Cantor for the acceptance of the infinite in • mathematical practice , in some respects more so . (But: less “drama”.) Three aspects and, thus, parts of my talk: • PART I: Dedekind’s contributions to the rise of set theory PART II: His use of infinite sets in mathematics more generally PART III: The issue of “explanation” as part of “mathematical practice”
3 Introduction (continued) Relevant works by Dedekind: 1872: Continuity and Irrational Numbers 1888: The Nature and Meaning of Numbers (Cf. 1930/32: Gesammelte Mathematische Werke, Vols. I-III ) But also: 1872-1899: Some meetings, and intermittent correspondence, with Cantor 1860s-90s: Supplements to Dirichlet’s Lectures on Number Theory ; also his corresponding work (with H. Weber) in algebraic geometry (function fields) 1855-1858: Early work on algebra, including Galois theory (lecture notes). Besides Zermelo’s and Hilbert’s remarks, I am building on the following: José Ferreirós: Labyrinth of Thought (1999/2010) • Akihiro Kanamori: “In Praise of Replacement” (BSL, 2012) • But also, recent work on mathematical explanation (cf. my “Dedekind, • Structural Reasoning, and Mathematical Understanding”, 2009) As more general background, cf. my survey “Dedekind’s Contributions to • the Foundations of Mathematics” (SEP, 2008/2011)
4 PART I: Dedekind’s Contributions to Set Theory (Reminder) • 1872 booklet (on continuity and irrational numbers): ▫ He starts with Q , seen as an infinite set—actual infinity, contra Aristotle ▫ Use of cuts (infinite subsets of Q ) to define continuity and introduce the elements of R • 1888 booklet (but much already in early drafts from 1870s): ▫ Use of a general theory of sets (“Systeme”) and functions (“Abbildungen”), both understood extensionally, as a foundational framework—for N , then for Z , Q , R etc. ▫ Explicit definition of sets as (Dedekind-)”infinite”—very bold, turning a “paradox” (Galilei) into a definition (characteristic property of infinite sets). (Here also: implicit use of AC.) ▫ Dedekind-Peano axioms for N , via “simply infinite systems” — acknowledged by Peano. ▫ Systematic justification of definitions by recursion and proofs by induction, via the notion of “chain” (a set closed under a given function)—later generalized by Zermelo and von Neumann. ▫ Famous categoricity result for simply infinite sets (all isomorphic). ▫ The “construction” of a simply infinite set (cf. Bolzano, put in problematic “psychologistic” language)—later the acknowledged basis for Zermelo’s axiom of infinity. • Correspondence with Cantor (from 1872 on): ▫ Proof that the set of algebraic numbers, not just Q , is countable—part of the inspiration for Cantor’s study of the cardinality of R (non-countability discovered in 1873). ▫ Proof of the Cantor-Bernstein equivalence theorem, again via Dedekind’s theory of chains. All of this became a standard and integral part of ZFC—thus Zermelo’s remark. Then again: Use of a naïve approach to sets, subject to Russell’s antinomy; and no basic axioms formulated explicitly (cf. Frege’s criticism, Zermelo’s work).
5 PART II: Dedekind’s Novel Uses of Infinity (More Generally) • In the more foundational works (1970s-80s): ▫ Q, then also R and N , as infinite sets—or rather, as infinite relational systems (ordered etc.). ▫ Real numbers as infinite sets of rationals (implicit use of, essentially, the power set axiom). ▫ Construction of Z , Q in terms of infinite equivalence classes or pairs in Dedekind’s Nachlass . • In algebraic number theory (from 1870s on): ▫ Arbitrary sub-fields, as well as corresponding sub-rings, of C . ▫ Ideals introduced as infinite sets (subsets of rings closed under certain operations). ▫ Similarly for other (often infinite) relational systems, e.g., modules, later lattices, etc. • In algebra (already in the 1850s): ▫ Quotient constructions for modular arithmetic: actually infinite residue classes treated as unitary mathematical objects here (unlike, e.g., in Gauss who works with residues directly). ▫ Important: Z [ x ], the ring of polynomials with integer coefficients (whose roots are algebraic numbers). Mod p : a class consisting of infinitely many infinite equivalence classes. ▫ In Dedekind’s own words: “[T]he whole system of infinitely many functions of a variable congruent to each other modulo p behaves here like a single concrete number in number theory. […] The system of infinitely many incongruent classes—infinitely many, since the degree may grow indefinitely—corresponds to the series of whole numbers in number theory.” (Dedekind 1930/32, Vol. 1, pp. 46-47, as quoted in Kanamori 2012, p. 49.)
6 PART II: Dedekind’s Uses of Infinity (Further Analyzed) • Basic observations: ▫ In some of these constructions, one can avoid the actual infinite easily (cf. Z , Q , also Z p ). ▫ But in other cases, the use of the actual infinite is unavoidable and essential—e.g., real numbers as cuts, ideals as infinite sets, and certain groups. ▫ Thus Kanamori’s remark (concerning the case Z [ x ]): “One can arguably date the entry of the actual infinite into mathematics here [i.e., in the 1850s], in the sense of infinite totalities serving as unitary objects within an infinite mathematical system” (pp. 49-50). • Towards “explanation”: ▫ In Dedekind’s corresponding writings, one can find very modern looking theorems, especially homomorphism and isomorphism theorems. (Example: Given a group homomorphism of G onto H , with kernel K , we have G / K ≅ H .) ▫ They are part of an emerging, very general methodology, where we study relational systems (finite or infinite sets with certain functions and relations defined on them) and various structure-preserving mappings between them. ▫ It is an infinitary, non-constructive, and “structuralist” methodology (cf. Reck 2009); often people talk about “abstract” mathematics in this connection (“abstract algebra” etc.). Both model theory and category theory are outgrowths of it. ▫ Here: not (always) an issue of “foundations”, but of “methodology”, “reasoning style”, etc.
7 PART III: Explanation and Mathematical Practice • Cantor: ▫ An alternative construction of R , via Cauchy sequences (related to Weierstrass etc.). ▫ Cantor’s theory of transfinite cardinal and ordinal numbers etc.; leading to the General Continuum Hypothesis and other questions central to higher set theory. ▫ Moreover, Cantor had a relatively sophisticated response to the set-theoretic antinomies. ▫ In addition, there was the beginning of descriptive set theory (point sets etc.). ▫ All of it was, at least initially, an outgrowth of Cantor’s work in analysis (the study of real- valued functions with infinitely many singularities)—connected to mainstream mathematics. • Dedekind: ▫ Important particular contributions to the rise of set theory as well, i.e., some central results. ▫ The systematic use of infinite sets, and corresponding functions, as a foundational framework , for studying N , Z , Q , R , C , recursive processes, sets more generally, etc. ▫ Beyond that: steps towards “abstract algebra”, studies in algebraic number theory, algebraic geometry, etc.—picked up later by Noether, Bourbaki, etc. • Thus: ▫ With respect to a foundational perspective, Dedekind was as important as Cantor. (Unlike Dedekind and Frege, Cantor was initially not very interested in foundational issues.) ▫ This is not to deny the importance of Cantor’s unique contributions (transfinite numbers, GCH, descriptive set theory, etc.), which had a huge influence on higher set theory . ▫ Then again, with respect to mainstream mathematics Dedekind’s influence may have been more pervasive than Cantor’s —and in ways that involve the infinite systematically.
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